Description
A set of distinct n positive integers is called a Diophantine n-tuple, if the product of any two distinct terms from the set increased by one is a square. Diophantine tuples are of ancient and modern interest,
with a huge literature.
In the present talk, extending the problem of Diophantine tuples, we study Diophantine graphs. Given a finite set V of positive integers, the induced Diophantine graph D(V) has vertex set V, and two numbers in V are linked by an edge if and only if their product increased by one is a square. A finite graph G is a Diophantine graph if it is isomorphic to D(V) for some V.
We present various results for Diophantine graphs, concerning representability and extendability questions, related to the edge density, and also for their chromatic number. To prove our results, we need to combine various tools, including congruences, (simultaneous) Pell-type equations, elliptic curves, bounds for various counting functions related to the number of distinct prime factors (e.g. of Hardy-Ramanujan type), and combinatorial and numerical methods.
The presented new results are joint with G. Batta and A. Pongracz.
Zoom: https://us06web.zoom.us/j/83970407396?pwd=BabVvtkl4paAHcx5lDo4Ar1XyP1Wvy.1
Meeting ID: 839 7040 7396
Passcode: 392793